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Sunday, March 9, 2008

A bunch of people have been Google searching for "How is Vega Calculated", and finding my post on Black-Scholes. I left out the calculation of vega, so I'll give the details.

Recall that a call price is a function

CP(S, X, t, v, r), where

S is the current stock priceX is the strike pricet is the time to expirationv is the volatility assumptionr is the interest rate assumption and assumption for the expected growth rate of the underlying

Vega is defined to be the partial derivative d(CP)/dv.

Typically, the function CP is calculated via a binomial tree expansion. For American options this is necessary, because of the early exercise right. For European options, there is an explicit formula you can differentiate.

To calculate vega, you do a numerical approximation of the derivative. Suppose epsilon = 0.001. In that case, vega is (CP(v + 0.001) - CP(v) / 0.001). You can't make epsilon too small, because floating point round-off will introduce error if epsilon is really small. A value of 0.001 is good enough.

To calculate vega, you construct two binomial trees, numerically approximating the derivative.

That's how you calculate vega. It's very simple. It's just numerically approximating the derivative.

Rho is calculated similarly. Rho is d(CP)/dr. You calculate two separate trees and numerically approximate the derivative.

Delta, gamma, and theta can be calculated directly from the binomial tree. Delta is d(CP)/dS. Delta is your correct hedge, which you calculated when you priced the option. Gamma is d(delta)/dS = d(CP^2)/(dS)^2. Gamma can be calculated by looking at delta at the root and delta at the first level of the tree. Theta is d(CP)/dt. Theta can also be read directly off the tree, by comparing the option price at the root with the prices at the first level of the tree. Alternatively, you could calculate a second binomial tree; some options traders think the latter method is better.

Based on my experience, these "sophisticated financial models" are usually a direct application of the compound interest formula and numerically approximating a derivative or integral. Really complicated financial models usually are complete nonsense.

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My Favorite Links

Here is a collection of my favorite links.

Personal Finance

For personal finance, my most frequently visited site is Yahoo Finance. Yahoo Finance has the best system for watching your stock quotes during the day. I also like the Motley Fool. Both of these websites encourage you to do independent thinking about finance.

My favorite discount online broker is Vanguard. They are not the cheapest commission-wise, but their customer service has been excellent. Plus, they give a high credit interest rate on the cash portion of your account.

Mises, Rothbard, and Austrian Economics

The school of "Austrian Economics" advocates credit-based money instead of debt-based money. There are two separate websites, www.mises.org and www.mises.net. These philosophies are a precursor to agorism. However, they still hold out false hope that the people who control the government can be convinced to switch to a fair monetary system. They fall short of the correct conclusion that government itself is the problem.

The Mises and Austrian school is still a pro-State theory of economics. They say "government should adopt a sound monetary policy instead of an unsound monetary policy". They fall short of the truth, which is "Who needs a government?"

Agorism and Anarcho-Capitalism

The primary source most commonly cited is agorism.info. Agorism.info has good introductory material, but I'm already looking for more advanced topics. I also found TOLFA interesting. The Molinari Institute has a lot of interesting links.

The source with the most advanced material on agorism is Kevin Carson's The Mutualist Blog.

This link on the History of Money has a lot of interesting bits on how bankers have controlled the world's money supply for hundreds of years or longer. Unlike most other sources, it is very short and to the point. However, their recommended solution falls short of true agorism.

Freedomain is another good read. He doesn't update his blog often, but he has a lot of good stuff posted in the past.

Kevin Carson's Mutualist Blog - This is a great source. He is tough to read at times, but his content is great. He's the best source on agorism I've seen. I like to take his topics and present them in simpler language. He updates his blog sporadically, but he has a lot of great content. It's also worth reading his other books and articles, which are available from his mutualist.org website. I also like the way Kevin Carson frequently links back to his favorite older posts. Kevin Carson's Shared Items is also worth reading; it's a list of posts from other blogs that he finds interesting.

Kung-Fu Monkey. This blog is written by someone who works as a writer in the entertainment industry, which explains the high quality of writing. He sounds like a closet agorist, although he hasn't specifically mentioned that philosophy. This post on the Extrapolated Everyday Bull**** Comparison has promoted Kung-Fu Monkey from my hitlist to my "read regularly" list.

Redpillguy's Blog - His blog is relatively new, so it's hard to judge. He doesn't really update his blog that often. On the other hand, he frequently cites my content, and that's certainly the sort of thing I appreciate.

Tranarchism is another new blog. It's too soon to judge the content. On the other hand, anyone who heavily cites my stuff can't be all bad. It's too infrequently updated.

Wally Conger's Blog is another good read. However, he really has two separate blogs mixed together. He has a lot of good stuff on agorism and libertarianism. However, he also likes to talk about his favorite movies and TV shows a lot.

Blog HitlistThere are blogs I'm currently evaluating to see if they're worth a regular read. I currently manage my hitlist through Google Reader.

Honorable Mention

These blogs have some interesting content, but they don't make it into my regular reading rotation. If they improved their content or improved their posting frequency, then they would be in my regular reading list. I check back occasionally, and on a slow day I might read them.

Bill Rempel - He talks about finance and trading. He really dislikes the Federal Reserve. I'm not sure if he's come all the way to agorism yet, but perhaps he can be coaxed. He's guilty of my #1 blog pet peeve: A PARTIAL RSS FEED!

Bored Zhwazi - Has some nice content, but it really isn't updated that often. It's worth checking back once every month or two.